TEST II Math 232 March 18 2004 Name By writing my name I swear by the honor code Read all of the following information before starting the exam Show all work clearly and in order You will not get full credit if I cannot see how you arrived at your answer even if your final answer is correct Make sure that you follow the directions in each problem and that your answer matches what is asked for Justify your answers algebraically whenever possible For most problems work done by calculator will not receive any points although you may use your calculator to check your answers Please keep your written answers brief be clear and to the point I will take points off for rambling and for incorrect or irrelevant statements By writing your name above you agree to the JMU honor code In particular this means that you may not use any notes or crib sheets during this exam that all work must be your own and that you may not obtain advance information revealing the problems on this exam This test has 8 problems and is worth 100 points plus some extra credit at the end Make sure that you have all of the pages Good luck 1 12 pts Determine whether each of the following statements is true T or false F a T F For all real numbers x we have 1 cot2 x csc2 x b T F If is an integer multiple of then cos 1 c T F The period of f x tan x is d T F If sec x 2 then cos 1 x 21 e T F A limit of the form is in an indeterminate form f T F 2 A limit of the form 0 is in an indeterminate form 10 pts Give short answers a Give a formal definition of sin for any angle Your definition should include the words unit circle standard position terminal edge and coordinate You can illustrate your definition with a picture if it makes it clearer b Sketch a clear labeled graph of f x sec 1 x It may help to begin by sketching the graph of the restricted secant function if you do this make sure it is clear which graph is which 3 10 pts Prove that d dx ln x 4 10 pts Prove that d dx cot x x1 Hint Start with the fact that eln x x for all x 0 csc2 x Hint Use the derivatives of sin x and cos x 5 24 pts Fill in the blanks a lim log3 x x 0 b d ln x dx c If lim ln f x then lim f x d If is an angle in standard position whose terminal edge intersects the unit circle 1 at the point 4 415 then tan e f g h i j x 0 x 0 sin x x ex lim lim ex sin x x lim x 0 x 1 cos x sin x2 4 x 2 x2 4 lim lim arctan x x d sec 1 x dx k Suppose f x is a general sine function with period 4 and a minimum at x 2 The first maximum to the right of this minimum occurs at x l Two points for writing anything you want plus two extra points for writing the secret word from the policy sheet 6 10 pts Use logarithmic differentiation to find the derivative of f x xsin x Show your work very carefully and circle your final answer 7 24 pts Fill in the blanks a The domain of f x csc x is b The range of f x csc x is c The domain of f x cos 1 x is d The range of f x cos 1 x is e sin sin 1 x x for all x in the interval f The graph of the function f x 4 sin 3 x 2 has amplitude period center point g If sin 1 x and sin is negative then is in the h List all the x values of the inflection points of f x cos x quadrant Survey Questions 2 extra credit points Name a question or topic that could have been on this test but wasn t How do you think you did SPACE FOR SCRAP WORK